Suwanee does have a bit of a town center, with a big lawn/amphitheater where they host outdoor events. It's a destination they built, not anything that came about organically by people living around it (I'd bet they're encouraging nearby housing now though). They have movie nights, regular food truck events, fun runs, concerts, etc. But it's not the compact, walk to every kind of store possible, laid out in neat square blocks kind of "town" that I think most people (who live in true cities) envision when they start talking about town centers. It's in a really nice area though, and like I said, they hold a number of community events throughout the year.
I'm sure you've seen the official website, but just in case:
http://www.suwanee.com/It is in Gwinnett county (note there are 2 n's in Gwinnett and only 1 in Suwanee), which is pretty large. That can be a pro and a con. It's one of the largest school systems in the US by enrollment (award-winning but
very large enrollments per school), and it has an extensive and award-winning parks system. Seriously, there are a lot of parks. They're not like the national parks with major hiking trails, etc., but they're still very nice outdoor spaces with volleyball courts, lakes, running paths, sports fields, and a variety of other activities. There are several national parks within an hour's drive if that's more what you want, though. Suwanee also puts an emphasis on preserving green space within the city itself.
Several minor league teams for professional sports are located in Gwinnett, including baseball, hockey, and soccer teams. The Gwinnett Arena seems to have almost as many big-name concerts as Phillips Arena in Atlanta proper, and the Gwinnett Civic Center hosts performing arts shows, expos, conventions, etc. You'd be close to most of those venues, should you be interested in those sorts of things (not the soccer one, though).
Traffic, if you are trying to get much further than Suwanee itself, is going to be a nightmare around rush hours IMO. The area is growing in popularity but there aren't many major roads to service it. Certain segments of roads might be more bike-friendly than surrounding areas, but I think you'd be taking a real risk to bike very far on any of the main thoroughfares because of the traffic volume and road characteristics. There are some bike paths, but I'm not sure how much they connect useful places. I think they're intended more for recreation/enjoyment than commuting or running errands. For public transportation, there is 1 local bus route that connects Suwanee to other major "destinations" (2 malls and the aforementioned Gwinnett Arena/Civic Center).
Culturally, I wouldn't necessarily call it progressive. I wouldn't say that it was stiflingly conservative, either. It's definitely right of center.
Also, if you do move there, make sure you're pronouncing it correctly: Swah-knee (with a bit more emphasis on the "ah" than the "knee"). Sue-wa-knee and its slight variations are a dead giveaway that you're not from around here. :) There's nothing wrong with that, of course. People just might look at you funny until they figure out what you mean (and then chuckle).
Note: My knowledge comes from living nearby my whole life (thought not quite as close as HairyUpperLip currently lives), driving through the area, and spending time with students/families who live in Suwanee.